Families torsion and Morse functions (Q2761081)

The R-torsion is a topological invariant introduced by Reidemeister-Franz for manifolds equipped with a unitarily flat vector bundle. Later, Ray-Singer defined the analytic torsion, which is an analytic version of the R-torsion, and proved that both of these torsions are equal for lens spaces [\textit{D. B. Ray} and \textit{I. M. Singer}, Adv. Math. 7, 145--210 (1971; Zbl 0239.58014)]. They also conjectured that this equality holds in general, which was later proven by \textit{J. Cheeger} [Ann. Math. (2) 109, 259--322 (1979; Zbl 0412.58026)] and \textit{W. Müller} [Adv. Math. 28, No. 3, 233--305 (1978; Zbl 0395.57011)]. Further generalizations of this conjecture were obtained by \textit{J. Lott} and \textit{M. Rothenberg} [J. Differ. Geom. 34, No. 2, 431--481 (1991; Zbl 0744.57021)] and \textit{W. Lück} [J. Differ. Geom. 37, No. 2, 263--322 (1993; Zbl 0792.53025)] for isometric group actions, and by \textit{J.-M. Bismut} and \textit{W. Zhang} for non-unitarily flat bundles [Astérisque 205 (1992; Zbl 0781.58039); Geom. Funct. Anal. 4, No. 2, 136--212 (1994; Zbl 0830.58030)].NEWLINENEWLINEThen Bismut-Lott constructed the higher analytic torsion for \(S^1\) fiber bundles equipped with a complex Hermitian line bundle with a unitarily flat connection, whose holonomy along the fibers is a root of unity [\textit{J.-M. Bismut} and \textit{J. Lott}, J. Am. Math. Soc. 8, No. 2, 291--363 (1995; Zbl 0837.58028)]; the zero degree component of this higher analytic torsion is given by the usual analytic torsion. On the other hand, by using algebraic \(K\)-theory, the higher R-torsion was defined by \textit{K. Igusa} [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 643--651 (1991; Zbl 0755.58016)] and \textit{J. R. Klein} [Proceedings of workshops held June 24--28, 1991, in Göttingen, Germany, and August 6--10, 1991, in Seattle, WA (USA), Contemp. Math. 150, 195--204 (1993; Zbl 0790.19006)]. Both of these higher torsions coincide for \(S^1\) fiber bundles over \(S^2\), but a general comparison formula seems to be very difficult to obtain.NEWLINENEWLINEIn this paper, the authors prove many properties of the higher analytic torsion of Bismut-Lott: it is extended to the equivariant setting, a proper normalization is given, rigidity formulas are proven, it is evaluated (modulo coboundaries) for families of manifolds with a fiberwise Morse function, and a formula is given for the case of unit sphere bundles. This generalizes the results of Cheeger, Müller, Lott-Rothenberg and Bismut-Zhang on the relation between analytic torsion and R-torsion, as well as computations by \textit{U. Bunke} for sphere bundles [Am. J. Math. 122, 377--401 (2000; Zbl 0984.58021)].